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1answer
128 views

Unipotent conjugacy classes

Consider a connected reductive group G over the complex numbers. Is there a `simple' formula for the number of conjugacy classes of unipotent elements in G?
0
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1answer
59 views

Decomposition of Lie subspaces

If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$ where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and ...
-1
votes
1answer
193 views

Decomposition of $S^7=Spin(7)/G_2$

The seven sphere can be written as the reductive space $S^7=Spin(7)/G_2$. Has the decomposition $Spin(7)=G_2\times S^7$ been calculated somewhere; maybe in terms of Cayley numbers?
4
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0answers
126 views

Parahorics and their normalizers

Let $G$ be a reductive group over a local non-arch field $F$. For convenience let's assume $G$ has anisotropic center (or even that $G$ is semisimple if preferred). Let $x$ be any point in the ...
1
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1answer
131 views

$\Gamma$-action on maximal tori in Borel-Tits

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...
3
votes
1answer
161 views

Weyl group action on continuous characters of the group of $\mathbf{Q}_p$-points of the torus

Let $G$ be a split reductive group over $\mathbf{Q}_p$ and assume $G$ has connected center. Let $T$ be a maximal split subtorus of $G$ and $R$ be the roots of $(G,T)$. Let $\chi : T(\mathbf{Q}_p) \to ...
1
vote
0answers
74 views

Splitting for Subsequence of Automorphism Sequence for Algebraic Groups

Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups $$ 1\to \mathrm{Inn}(G)\to ...
6
votes
0answers
118 views

Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
1
vote
1answer
63 views

Strictly contracting elements in the center of a Levi subgroup

Let $G$ be a connected reductive group over a non archimedean local field $k$. Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z_M \subset L$ be the center of $M$ and $S_M ...
1
vote
0answers
108 views

Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...
3
votes
1answer
106 views

Supercuspidal with Iwahori fixed vector

Let $F$ be a local field. Is there a reference for the following fact: No supercuspidal representation of $GL_2(F)$ has an Iwahori-fixed vector? I have a proof, by I'd prefer a reference, ...
0
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1answer
59 views

Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$, $g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$, $g$ is ...
3
votes
1answer
190 views

What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field. Here, I think the word "compact" is used ...
1
vote
1answer
128 views

Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field), $g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...
0
votes
1answer
160 views

number of simple representations

For a linearly reductive Group $G$ over a field $k$ one has that the category of finite dimensional representations of $G$ is semisimple. What can one say about the number of simple representations? ...
1
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0answers
126 views

Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field. Then $G(F)$ is a p-adic group. Let $\Psi(G)$ be the lattice of algebraic characters. Let $\Lambda_G$ be the ...
7
votes
1answer
369 views

Why people usually consider reductive groups in GIT?

Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT? I found that the property of completely reducible of ...
6
votes
2answers
256 views

Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...
7
votes
1answer
208 views

Open cell decomposition after applying a Weyl group element

Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step. For ...
2
votes
0answers
79 views

Invariant vectors in supercuspidal representations of GL_2(Zp)

Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the ...
3
votes
1answer
121 views

Semisimple group not split by an unramified extension?

Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?
6
votes
2answers
377 views

Reference request: expository text on the structure of reductive groups over non-archimedean local fields

I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner ...
3
votes
0answers
136 views

How to think about non-connected reductive groups

Suppose someone knows well the theory of connected reductive groups, over an algebraically closed field or more generally over any field, say for instance most of the content of Borel-Tits. Is ...
4
votes
1answer
261 views

How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups So let me phrase a precise question: Is there a standard technique by which one can translate the ...
5
votes
2answers
200 views

Principal series of finite group of Lie type

I have a naive question on complex representations of finite groups of Lie type. Let $\bf G$ be a reductive group (say connected, with connected center, for safety) defined over a finite field ...
3
votes
0answers
129 views

Decomposition of k-split tori of p-adic reductive groups

Let $G$ be a reductive group over a $p$-adic field $k$, $S \subset G$ a maximal $k$-split torus, $\Phi(G,S)$ the relative root system and $\Delta$ a basis of $\Phi$. There is a group homomorphism : ...
2
votes
1answer
230 views

Weyl group of the restriction of scalars of split reductive group

Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$. Set $G' = ...
4
votes
1answer
175 views

Steinberg reps of reductive groups over local fields vs finite fields

Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$. Edit: The statements only make sense modulo tensoring by one-dimensional representations. Are the unitary, ...
2
votes
2answers
190 views

a conjugacy question in quasi-split reductive groups

I have a somewhat technical question about conjugacy in quasi-reductive groups. Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected quasi-split reductive group over ...
21
votes
1answer
1k views

Reconciling Lusztig's results with the Langlands philosophy

Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = ...
20
votes
1answer
642 views

What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...
3
votes
1answer
219 views

Volume of PGL(2,F) \ PGL(2, A)

Let $F$ be a global field. What is the measure of $PGL_2(F) \backslash PGL_2(\mathbb{A})$? This depends of course on the normalizations of the Haar measures on $PGL_2(F)$ and $PGL_2(\mathbb{A})$. ...
2
votes
0answers
134 views

Do there exist pseudo-reductive (but not reductive) groups of small dimension?

I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...
10
votes
4answers
835 views

Is the normalizer of a reductive subgroup reductive?

Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the ...
4
votes
1answer
258 views

Does there exist a categorical treatment of root data(systems)?

What I am looking for is an abstract description of root data with their morphisms(!) plus a comparison with the categories of reductive groups over some field, Dynkin diagrams, Lie algebras, ...
4
votes
3answers
305 views

Are all irreducible supercuspidal representation induced from compact-mod-center subgroups?

Let $G$ be a reductive group over a local non-archimedean field $F$. Can every irreducible supercuspidal representation of $G(F)$ be realized as the induction from an open subgroup, which is compact ...
2
votes
2answers
341 views

description of an endomorphism algebra

Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus). I ...
7
votes
2answers
485 views

commuting elements in a reductive group

Does anyone know if the following holds? Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G. To make things easier, you ...
2
votes
1answer
234 views

Pseudo coefficients and orbital integrals

I am looking for a reference/idea, how this passage from Labesse's Snowbird Lecture "Introduction to endoscopy" pg.5 can be explained: "We shall denote by $f_\pi$ a pseudo-coefficient for $\pi$, ...
1
vote
1answer
203 views

Heights in reductive groups

Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...
3
votes
1answer
159 views

homogenous bundles

Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...
1
vote
1answer
389 views

moduli problem for flag varieties?

Hi, Suppose $G$ is a reductive group over an algebraiclly closed field $k$ (suppose $k$ of char zero if you want at first). Let $X$ be its flag variety. Question: What is the moduli problem that $X$ ...
1
vote
1answer
227 views

Intertwining Integral defined on a Weyl group?

Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group? ...
2
votes
1answer
420 views

When is compact induction in GL(2) from an open compact group admissible?

Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional ...
2
votes
1answer
212 views

Abel transform is an * isomorphism for SL(2, R)

Assume we conisder $G= SL(2, R)$, $K=SO(2)$ and $N$ the strict upper triangular matrices in $G$, $A$ diagonal matrices, and the Borel supgroup $B=NA$, $W$ Weyl group. Then we have an isomorphism of ...
3
votes
1answer
354 views

Canonical rational form for $SL(n)$

The canonical rational form helps us to parametrize the conjugacy classes in $GL(n)$ over any commutative field. How can we parametriize the conjugacy classes in $SL_n(k)$, where $k$ is an ...
2
votes
1answer
220 views

Representation theory of G1 versus G/Z

Let $G$ be an locally compact group $G$, then every irreucible representations $\pi$ is isomorphic to $\omega_{\pi} \otimes \pi'$, where $\omega_{\pi}$ is the central character of $\pi$ and $\pi'$ an ...
1
vote
2answers
171 views

What is this measure on the affine Weyl group?

Let $G$ be $SL(n, F)$ for a non-archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$. What are the properties of the function $w\mapsto \mu(IwI)/\mu(I)$ for ...
7
votes
2answers
558 views

Conjugation in GL(n) (p-adic setting)

In $GL(n, \mathbb{Q}_p)$, what are the orbits under conjugation of $GL(n, \mathbb{Z}_p)$?
5
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2answers
638 views

Parabolic induction GL(n,Zp)

Let $P$ be a parabolic subgroup of $GL(n)$ with Levi decomposition $P =MN$, where $N$ is the unipotent radical. Let $\pi$ be an irreducible representation of $M(\mathbf{Z}_p)$ inflated to ...